Heavy-traffic limit of a queue ($\epsilon Q_{\epsilon }(\lfloor t/{\epsilon }^2\rfloor )\Rightarrow \Gamma (\sigma B(t)-dt)$)

For fixed independent and identically distributed arrivals of mean $\mu$ and positive finite variance ${\sigma }^2$, put service at $\mu +\epsilon d$. Rescaling net input by space $\epsilon$ and time ${\epsilon }^{-2}$ gives Brownian motion with drift $-d$ by the functional central limit theorem. The continuous Skorokhod reflection map on the half-line gives the displayed transient limit, with a correspondingly scaled initial workload. A stationary approximation additionally requires justification of limit interchange. When valid, the stationary law of negatively drifted reflected Brownian motion predicts $\mathbb{P}(Q_{\epsilon }>b/\epsilon )\approx e^{-2db/{\sigma }^2}$.